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Question 4 Let V denote the real vector space of polynomials of degree at most 4. Define an inner product on V as follows: For polynomials f (x). g(x) EV, (f ( x ), g (x)) = [ f(x)g(x) dx. Let w,(x) = 1+ x*,w2(x) = 1+ x , wa(x) = x3, let B = { w, (x),w2(x),w}(x) }, and let W be the subspace of V spanned by B. (a) Show that B is a linearly independent set in V. ) (b) Use the Gram-Schmidt orthogonalization procedure on the ordered basis B to find an orthogonal basis C of W, and hence, determine an orthonormal basis D of W. (c) Find a basis of WJ, the orthogonal complement of W in V. (d) Define a linear operator T: W - W be defined as follows: T(W] (x)) = WI(x) + w2(x) + w=(x) T(W2(x) ) = WI (x) -W2(x) T(W3 (x)) = W(x) + w2(x) -2w3(x) Compute e, the matrix representation of T with respect to the orthogonal basis C found in part (b)